Mixing trichotomy for an Ehrenfest urn with impurities

We consider a version of the classical Ehrenfest urn model with two urns and two types of balls: regular and heavy. Each ball is selected independently according to a Poisson process having rate 1 for regular balls and rate alpha is an element of (0, 1) for heavy balls, and once a ball is selected, is placed in a urn uniformly at random. We study the asymptotic behavior when the total number of balls, N, goes to infinity, and the number of heavy ball is set to mN is an element of {1, ... , N - 1}. We focus on the observable given by the total number of balls in the left urn, which converges to a binomial distribution of parameter 1/2, regardless of the choice of the two parameters, alpha and mN. We study the speed of convergence and show that this can exhibit three different phenomenologies depending on the choice of the two parameters of the model.